Regularizations involving the L\(_1\)  or L\(_p\) -norm of the solution are used under the assumption that the optical properties (representing the image) to be reconstructed are spatially sparse. These regularizations tend to oversparsify the distribution of the optical properties when such an assumption does not hold [1]}, for example, in the case of multiple activations or complex injuries in the brain, where the features of interest are not spatially localized and the optical properties relative to the background are therefore non-sparse [2]}. In order to be able to reconstruct images in which edges are preserved and features are not spatially sparse, a different approach is required. Total variation (TV) regularization, which uses the L\(_1\) -norm of the gradient of the solution as a regularizing term (the detailed forms will be given in Section REF  and REF ), can be used to overcome the limitations. The gradient operator can transform the solution \(\mu ^*\)  to a sparse space where non-zero values only occur at sharp features. As such, TV can perform better than the pure sparsity preserving regularizations at preserving edges of objects in the images that are not sparse. Further, gradient is a highpass operator, which imposes smoothness to the solution. This improves the conditioning of the minimization problem (Eq. (REF )), thus enabling a robust numerical solution. Due to these advantages, TV has been adapted from applications in imaging processing [3]}, [4]}, [5]} to various medical image reconstruction problems, including photoacoustic tomography (PAT) [6]}, bioluminescence tomography (BLT) [7]}, fluorescence tomography (FT) [8]}, as well as DOT [9]}, [10]}, [11]}.
